Infinite summation portfolio. A series is a sum of terms of a sequence. A finite series, has its first and the last term defined, and the infinite series, or in other words infinite summation

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International Baccalaureate

 Math Standard Level Internal Assessment

Portfolio Type: I

Portfolio Title: Infinite Summation

Due date: 9th of December, 2011

Teacher: Mr. Peter Vassilev

School: The American College of Sofia

Candidate Name: Rami Meziad

Candidate number: 002368-008

Examination Session: May 2012

Introduction:

A series is a sum of terms of a sequence. A finite series, has its first and the last term defined, and the infinite series, or in other words infinite summation  is a series which continues indefinitely. The Taylor's theorem  and the Euler-Maclaurin's formula  will help us solve our given infinite summation, which is: ,,,,

And by adding different values for x and a, we will be able to find a general pattern in which the sequences tends to move with. And this is mainly what this portfolio will ask us to do.

Method:

For our sequence, which is:

, we have to substitute in the case where x = 1 and a = 2. After that, we have to calculate the first n terms which happen to be eleven to fulfill the given condition.

So after substitution we get

Now let's calculate for n, when :

In fact, t9 and t10 are not equal to 0, but since we have to take our answers correct to six decimal places, we can't see the real values. However, the numbers become so small, that they become insignificant, or in other words they are equal to 0.

Now, we need to find the sum of Sn :

Now, using Excel 2010, let's plot the relation between Sn and n :

Looking at the graph, we can notice that Sn increases rapidly at first, and then it evens out when it reaches 2, which seems like an asymptote. The same happens with the terms’ values. They decrease rapidly until they reach the 0, which if we plot will seem like its asymptote.

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Therefore, we can see that both move a maximum of 1 unit away from their first point, and then even out to the mentioned asymptote.

For Sn , the asymptote is .

For the terms’ calculation for given n, the asymptote is .

Therefore:

As n approaches infinity, Sn approaches 2:

,

Now, we do the same thing as before, but for  with the same condition for n ():

             

Now we have to calculate Sn again, but for  :

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